Hochschild cohomology

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\underline{Hochschild cohomology}
 
Hochschild cohomology is tied up with the very basics of
$A_\infty$-structures,
but it also plays a more direct role in Fukaya categories through ``bulk
deformations''. This talk has two kinds of prerequisites: one is the
knowledge of 
Hochschild cohomology at least as it appears in the deformation theory of
algebras, 
and the other one is some knowledge of Fukaya categories. There are few
references apart
from the inevitable [Fukaya-Oh-Ohta-Ono, book] and the papers by the same
authors on toric
varieties.
 
\underline{Plan}
 
{\it Definition.} We start with the Hochschild complex of an ordinary
graded algebra,
and the Lie bracket on that complex. There are many references, such as
the book
[Hazewinkel et al, eds., Deformation of algebras and structures and
applications].
The $A_\infty$-structure equation appears as Maurer-Cartan equation in the
resulting
dg Lie algebra, which is relevant for deformation theory. In particular,
it explains a 
phenomenon called intrinsic formality: under suitable vanishing conditions
on the 
Hochschild cohomology of an algebra, every $A_\infty$-deformation of that
algebra
is isomorphic to the trivial one. You'll find a discussion in
[Seidel-Thomas,
braid group actions on derived categories of coherent sheaves].
 
As an example, the Hochschild-Kostant-Rosenberg theorem on the Hochschild
cohomology of polynomial algebras can serve. It also illustrates the two
structures
present on Hochschild cohomology (Lie bracket and product).
 
{\it Generalization.} More generally, one can consider the Hochschild
cohomology
of an $A_\infty$-algebra. This is actually obvious from the Maurer-Cartan
framework
above. Less obvious is the fact that we still have the commutative product
(that becomes 
straightforward if one thinks in terms of bimodules, but we'd problably
best avoid
that). Hochschild cohomology still governs formal deformations of the
$A_\infty$-structure.
 
{\it Application.} For any symplectic manifold $M$, there is a ring
homomorphism
\[
QH^*(M) \longrightarrow HH^*(\mathcal F,\mathcal F)
\]
from the quantum cohomology ring to the Hochschild cohomology of the
Fukaya category.
This is defined by considering discs with an interior marked point, and
one can
interpret it in deformation theory terms: (generalized) infinitesimal
deformations 
of the symplectic form induce deformations of the Fukaya category.
 
It is instructive to see an informal (picture) proof of the fact that the
map above is 
compatible with the ring structure. 
 
\underline{Notes}
 
For attempts to generalize the above homomorphism to other contexts, see
[Seidel, Fukaya
categories and deformations] (non-compact manifolds; somewhat
unsuccessful), and
[Seidel, More about vanishing cycles and mutations].
 
\underline{Dependencies}
 
Depends on having defined the Fukaya category, and also I think on the
talk about
quantum cohomology.
 
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